Hello all! Having worked for a while with various computer systems, primarily Active Directory and Exchange, I wanted to share some of my experiences with two objectives in mind: 1) obtain feedback to improve my mastery of those systems and 2) help others working on the same subject. Other posts are about CentOS, Citrix NetScaler, and VMware.
NOTE: most of my posts are in English but some others in French, with a summary in English. However, some of the CentOS blog posts lack this summary.

Problem 1 was caused by the filtered synchronization for which we opted in part 4.x.

Only users, contacts and groups in the following containers were synchronized with Windows Azure Active Directory:

Therefore, Windows Azure Active Directory had no reference to those users and cannot resolve their email address. The error message stated:

"The email address you entered couldn't be found. Please check the recipient's email address and try to resend the message."

The solution is to either synchronize all Active Directory containers holding objects likely to send and receive email, or simply move the objects in question to one of the OUs being synchronized with Office 365 (Windows Azure AD).

I chose the second option. Once this was completed, all members of the organization, both migrated and non-migrated, could communicate by email.

Problem 2 (and 3)

Problems 2 and 3 are unlikely to be encountered in a production environment but possibly in a practice network. My practice network accesses the Internet via an IP address dynamically assigned by my ISP. Inbound mail flow is achieved with a product called "No-IP" that dynamically adjusts DNS records, the MX records in particular, as the DHCP assigned IP addresses change.

However, many spam filtering services, such as SpamHaus, block email sent from dynamically assigned IP addresses.

This problem was resolved, at least temporarily, by a request sent to Microsoft tech support to unblock the IP address or whitelist the domain name. At this time, it is not clear which option was selected since the IP address is still the same. If the IP address changes (as it will) and mail still flows from point to point, we could assume that the domain name was whitelisted.

In any event, at the time of this writing, mail flow is successful in all directions and between all recipients.